Theorem rnun 5039

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5036	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4830	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4836	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2198	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4639	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4639	. . 3 |- ran A = dom `' A
7		df-rn 4639	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3289	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2208	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1353   u. cun 3129   `'ccnv 4627   dom cdm 4628   ran crn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  imaundi  5043  imaundir  5044  rnpropg  5110  fun  5390  foun  5482  fpr  5700  fprg  5701  sbthlemi6  6963  exmidfodomrlemim  7202